Propositional calculus is a branch of logic. It is called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic, it deals with propositions and argument flow. Compound propositions are formed by connecting propositions by logical connectives; the propositions without logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order higher-order logic. Logical connectives are found in natural languages. In English for example, some examples are "and", "or", "not" and "if"; the following is an example of a simple inference within the scope of propositional logic: Premise 1: If it's raining it's cloudy. Premise 2: It's raining. Conclusion: It's cloudy. Both premises and the conclusion are propositions.
The premises are taken for granted and with the application of modus ponens the conclusion follows. As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements: Premise 1: P → Q Premise 2: P Conclusion: Q The same can be stated succinctly in the following way: P → Q, P ⊢ Q When P is interpreted as "It's raining" and Q as "it's cloudy" the above symbolic expressions can be seen to correspond with the original expression in natural language. Not only that, but they will correspond with any other inference of this form, which will be valid on the same basis that this inference is. Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of inference rules and axioms allows certain formulas to be derived.
These derived formulas may be interpreted to be true propositions. A constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem; the derivation may be interpreted as proof of the proposition represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly; the natural language propositions that arise when they're interpreted are outside the scope of the system, the relation between the formal system and its interpretation is outside the formal system itself. In classical truth-functional propositional logic, formulas are interpreted as having one of two possible truth values, the truth value of true or the truth value of false; the principle of bivalence and the law of excluded middle are upheld. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be zeroth-order logic. However, alternative propositional logics are possible.
See Other logical calculi below. Although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics; the logic was focused on propositions. This advancement was different from the traditional syllogistic logic, focused on terms; however in antiquity, the propositional logic developed by the Stoics was no longer understood. The system was reinvented by Peter Abelard in the 12th century. Propositional logic was refined using symbolic logic; the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic was an advancement from the earlier propositional logic.
One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." Predicate logic ushered in a new era in logic's history. Natural deduction was invented by Jan Łukasiewicz. Truth-Trees were invented by Evert Willem Beth; the invention of truth-tables, however, is of uncertain attribution. Within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables; the actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post. Besides Frege and Russell, others credited with having ideas preceding truth-tables include Philo, Charles Sanders Peirce, Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Ernst Schröder, Alfred North Whitehead, William Stanley Jevons, John Venn, Clarence Irving Lewis; some have concluded, like John Shosky, that "It is far from clea
Michael Moore is an American jazz musician who has lived in the Netherlands since 1982. The son of a semi-professional musician, Moore was raised in Eureka, California, he studied music at Humboldt State and in 1977 graduated from the New England Conservatory of Music, where he studied with Jaki Byard and Gunther Schuller, was a classmate of Marty Ehrlich. He played in a variety of musical contexts those in support of theatre and dance groups. By 1982 he was a regular member of Misha Mengelberg's Instant Composers Pool and had moved to Amsterdam, he was a member of Georg Gräwe's Grubenklang Orchester. Moore is one-third of the Clusone Trio with drummer Han Bennink. Meant only to play a single date at a festival in Clusone, the trio toured irregularly for several years and recorded six albums, including one of freely-interpreted Irving Berlin compositions. Moore's first recording as a leader was in 1992 but it was with 1994's Chicoutimi that he began to earn recognition as a composer; the drummerless trio on this album was inspired by the duo recordings of Lee Konitz and Gil Evans and recalls in places the Jimmy Giuffre trios of the early 1960s.
Moore plays in Jewels and Binoculars, a collective trio with bassist Lindsey Horner and drummer Michael Vatcher, devoted to interpretations of Bob Dylan songs. In 1986, Moore won the most prestigious jazz award in the Netherlands. In 1991, he founded Ramboy Records to document his music. Moore started his jazz quintet in 2005, in which he works with accomplished Dutch players: trumpeter Eric Vloeimans, pianist Marc van Roon, bassist Paul Berner, drummer Owen Hart, Jr. In October 2005, the quintet recorded the album Osiris. In 2013, he performed with InstanPool, a group of international musicians making improved music and playing a composition. InstanPool consists of Michael Moore and Mark Alban Lotz, Korhan Erel and Robert van Heumen, Sevket Akinci and Giray Gürkal; some members are in Islak Köpek. MGM Trio with Marilyn Crispell Thirteen Ways with Fred Hersch Monitor The Voice Is The Matter with Jodi Gilbert Mt. Olympus with Alex Maguire Pursuit with Benoit Delbecq 5 White Widow Air Street Floating 1...2...3 Jewels & Binoculars Kamosc with Achim Kaufmann Simple Songs with Celano Baggiani Group This We Know with Fred Hersch Fragile PPP5 with Cor Fuhler Live In NYC with Will Holhauser Easter Sunday Coconut with Eric Boeren Something Nothing with Achim Kaufmann Nothing Something with Achim Kaufmann Furthermore with Achim Kaufmann Felix Quartet With Dave Douglas Mountain Passages Biography at Ramboy Recordings Whitehead, Kevin.
New Dutch Swing. New York: Billboard Books. ISBN 0-8230-8334-9. Complete discography at Ramboy Recordings ICP Orchestra homepage
Kazuki Ōmori is a Japanese film director and screenwriter. Born in Osaka, Ōmori studied at Kyoto Prefectural University of Medicine and holds a license to practice medicine. While in school, he began making films independently, with Kuraku naru made matenai!, which featured Seijun Suzuki, receiving high praise. His script "Orenji rōdo kyūkō" won the 3rd Kido Award for screenplays in 1977, the next year he was able to film that in his professional debut. Several of his films, such as the 1980 Hipokuratesu-tachi, feature doctors or rely on his knowledge of medicine, he has worked in a variety of genres, including suspense films and most famously abroad, several contributions to the Heisei Godzilla series.Ōmori participated in the formation of Director's Company in 1982, an independent production company founded by nine directors, including Kiyoshi Kurosawa, Sōgo Ishii, Shinji Sōmai, Kazuhiko Hasegawa. In 2000, he became a professor at Osaka Electro-Communication University, in 2005, a professor at Osaka University of Arts.
He was a special guest at G-Fest XIII in 2006. Kuraku naru-made matenai! Orenji Rodo kyuko Disciples of Hippocrates Kaze no uta o kike Sukanpin walk Koisuru onnatachi Totto Channel Sayonara no onnatachi Hana no furu gogo Godzilla vs. Biollante Boku ga byoki ni natta wake Mangetsu: Mr. Moonlight Godzilla vs. King Ghidorah Keisho sakazuki Shoot Dai shitsuren Kinkyu yobidashi - Emajenshi koru Waga kokoro no ginga tetsudo: Miyazawa Kenji monogatari Dorimu sutajiamu June Bride The Boy Who Saw the Wind Hakata Movie: Chinchiromai Saiaku Hashire! Ichiro T. R. Y. Super Star Fleet Sazer-X the Movie: Fight! Star Soldiers Kuraku naru-made matenai! Orenji Rodo kyuko Kaze no uta o kike Hipokuratesu-tachi Take It Easy Koisuru onnatachi Totto Channel Sayonara no onnatachi Yojo no jidai Hana no furu gogo Godzilla vs. Biollante Boku ga byoki ni natta wake Mangetsu: Mr. Moonlight Godzilla vs. King Ghidorah Godzilla vs. Mothra Kinkyu yobidashi - Emajenshi koru Godzilla vs. Destoroyah June Bride Kaze o mita shonen Hakata Movie: Chinchiromai Hashire!